Asymptotic theory for conditional U-statistics on hyperspherical domains under measurement error

Abstract

We develop a deconvolution theory for conditional $U$-functionals of arbitrary order when the conditioning variable is a latent hyperspherical covariate observed under rotational measurement error. The statistical model assumes that the unobserved regressor $X\in S^d$ is observed only through $Z=UX$, where $U\in \mathrm{SO}(d+1)$ is an independent random rotation with known distribution. This gives rise to an inverse problem which is not covered by the standard Euclidean or compact-Lie-group deconvolution theories: although the noise acts through the rotation group, the predictor itself lies on the homogeneous space $S^d$, which is not a Lie group for $d\ne \mathrm{1,3}$. Consequently, the deconvolution analysis must be carried out through the interaction between spherical harmonics on $S^d$ and the rotational Fourier transform on $\mathrm{SO}(d+1)$. We introduce a spectral deconvolution estimator for general-order conditional $U$-statistics based on truncated Fourier–Laplace expansions and inversion of the rotational Fourier matrices associated with the measurement-error distribution. We prove uniform consistency of the induced product-density estimator and establish $L^2$-risk bounds for the resulting conditional $U$-functional estimator. The bounds quantify the combined effect of Laplace–Beltrami smoothness, harmonic multiplicities, the order of the $U$-statistic, and the ill-posedness of the rotational convolution operator. We further distinguish ordinary-smooth, super-smooth, and log-super-smooth rotational error regimes, thereby extending the classical smoothness taxonomy of deconvolution to hyperspherical domains through the spectral behavior of inverse rotational Fourier matrices. In the ordinary-smooth regime, we derive pointwise asymptotic normality by combining a Hoeffding decomposition with variance estimates for deconvolution-based hyperspherical $U$-statistics. This yields asymptotic confidence intervals for nonlinear conditional interaction functionals, including conditional Kendall-type dependence measures. The finite-sample behavior of the method is investigated through Monte Carlo experiments on $S^2$, and its applicability is illustrated on directional data. The results provide a general asymptotic framework for conditional $U$-statistics with noisy hyperspherical predictors beyond the directly observed and Euclidean settings.